Following the suggestion of one of the commenters, I introduced Katie to the concept of inverse functions. I started by saying that I want to tell her about two functions that are not friends with each other. She seemed interested, so I continued. “One function takes circles and converts them to squares and the other one takes squares and converts them to circles. And so they always quarrel: ‘Don’t convert my circles to squares!’ one says, ‘Don’t convert my squares to circles!’ replies the other.”
Immediately Katie started waving her hands at me and saying, “now I want to tell you about two functions.” “One function converts purple things to green things and the other one converts green things to purple things. And they always quarrel!” And she went on to mimic my functions quarreling, but only with her functions.
Later in the day Katie wanted to tell me about another function that she came up with: bad pirates go in and good people come out. When I asked her about the function that is not friends with this one she replied correctly, but she also said that she does not like that function :-).
We have since come up with a few more pairs, but I will not list all of them here. I did at some point use the exact example that was suggested in the comment, with a slight twist. There were two dwarfs: one liked to break toys and the other one would fix them. And naturally they did not get along.
Please keep your comments and suggestions coming!